How Wide is the Ocean?
Over the summer many of you will have had holidays at the seaside. If you are standing at the top of a cliff (not too near the edge!), how far away is the horizon? Here is a way of estimating that distance if you know the height of the cliff.
Suppose you are on a cliff H miles high, then we want to find D .
By Pythagoras’ theorem,
| (R + H)² | = | R² + H² |
|
| = | R² + 2RH + H² – R² |
|
| = | 2RH + H² |
D²But you probably know the height of the cliff in feet, so
| H | = |
|
And the radius of the earth, R, = 3960
|
| = |
| + |
|
The second term is very small and can be neglected so D² = 1½h
So if your cliff is 150 feet high, you can see 15 miles. A slightly rougher approximation is __ 1¼ h. If your information about the cliff is in metres,
D = 
then you can show that the corresponding formula for the distance of the horizon
| __ |
| ||
| in kilometres is about | 3½ | h. |
|
Martin Perkins
